Question

Show by example that Lemma 9.2 may be false if is not normal.

Short answer

It has been proved that Lemma 9.2 is false ifis not normal.

Step-by-step solution

Step-by-Step SolutionStep 1: State the lemma

Lemma 9.2: Let$M$ and$N$ be normal subgroups of a group$G$ such that$M\cap N=⟨e⟩$ . If$a\in M$ and$b\in N$ then,$ab=ba$ .

Take D as example

Consider the dihydral group $D_4=\{1,r,r^2,r^3,t,rt,r^{2}t,r^{3}t\}$

Here,$\left|r\right|=4$and $\left|t\right|=2$

This group follows the relation as:

$\begin{array}{c}rt=t{r}^{-1}\\ =t{r}^3\end{array}$

Consider its subgroups role="math" localid="1652783481702" $M=\{1,r,r^2,r^3\}$and $N=\{1,t\}$

Here,$M\cap N=\left\{e\right\}$but $rt\ne tr$for $r\in M,t\in N$as $rt=t{r}^{-1}$, $rt=t{r}^3$, $t{r}^{-1}=t{r}^3$and $r\ne r^3$.

Thus, it can be concluded that if$N$ is not normal then,$ab\ne ba$ .